54 «Dr. G. Green on Ship-Waves, and on Waves in 



§ 3. Fluid Motion due to Moving Submerged 

 Cylinder. 



As a first illustration of the above process we may write 

 down the first three terms of the solution for the case of an 

 infinite cylinder moving with velocity v in the direction of 

 sb positive, its axis being parallel to the y axis. The terms 

 referred to in (a) and (/>) of § 2 are readily obtained from 

 the well known solution for the fluid motion due to the 

 steady motion of a cylinder in an infinite liquid : — 



<j> = — a 2 v cos djr = — a 2 vx , jr 2 , .... (23) 



in which the coordinates are referred to an origin coinciding 

 with the instantaneous position of the axis of the cylinder. 

 The surface elevation, f, required by (2) to determine the 

 most important part of the pressure term referred to in (c) 

 of § 2 is more readily obtained from the corresponding 

 stream function, representing the fluid-motion relative to 

 the cylinder, 



^ = — v sin (r— a* jr) = — vz + a 2 vz/r 2 . 



At an infinite distance from the cylinder each stream-line is 

 practically horizontal and each particle traversing the line 

 retains its initial vertical coordinate. Hence taking the 

 particles in the plane at z—li above the cylinder, we have 

 ty-=z—vli as the constant value of the stream function for 

 the particles of fluid lying in the free surface. Accordingly 

 the equation to the surface as determined by the (a) term 

 alone is 



-^=z-h = a^/r\ .... (24) 



and, to the degree of accuracy we are aiming at, we may 

 replace z by h on the right. The surface elevation due to the 

 (a) and (b) terms together is then given by 



f = -2a 2 hl(sj 2 + 7t 2 ). ... (25) 



To the same order of accuracy, the pressure which must be 

 kept applied to the surface to maintain the motion repre- 

 sented by terms referred to in (a) and (b) of § 2 is given by 



p= - 2g P a 2 h/{a, 2 + h 2 ). . . (26) 



This pressure must maintain a constant relation to the 

 cylinder, that is, in (26) we may regard x as measured from 

 an origin in the surface vertically above the instantaneous 

 position of the axis of the moving cylinder. In order to 

 make use of (17) and (18), however, it is convenient to use 



